1549301742-The_Theory_of_Difference_Schemes__Samarskii

290 Difference Schemes for Elliptic Equations

The expression for (A-y, y) can be transformed without difficulties

0

in just the same way. Recalling now that c 1 < ko:o: < c 2 and (Ay, y) =

L!=l (Yxa, YxJa we obtain c 1 A < A < c 2 A, yielding inequalities (28).

Inequality (29) can be deduced from the above relations by virtue of the

0

estimate f; E <A< 6. E.

4.5 HIGHER-ACCURACY SCHEMES FOR POISSON'S EQUATION

In this section we consider higher-accuracy schemes for the Dirichlet prob-

lem ( 1) of Section 1 in a rectangle.

1. The statement of the Dirichlet difference problem providing a higher-
   order approxi1nation. On the basis of the "cross" scheme it is possible to
   construct a scheme with the error of approximation 0(1h1^4 ) or O(h^6 ) on a
   solution in the case of a square (cube) grid. In order to raise the order of
   approximation, we exploit the fact that u = u(x) is a solution of Poisson's
   equation

(1) 6.u = -.f(x).

Without loss of generality we n1ay restrict ourselves to the careful analysis

of the two-din1ensional case (p = 2) where

32u

Lo: u = -8 2 '

x 0:

by appeal to the difference operator

Acx u = ·u,,, •"O'"O ~ ,

assuming u = u( x) to possess all necessary derivatives. Then

(2)

' h2 h2

Au-Lu=-^1 L^2 u+-2 L^2 u+O(lhl^4 ).

12 1 12 2

From the equation L 1 1l + L2 u = -.f ( x) we find that

so that

(3)